67.24.4 problem 34.5 (d)
Internal
problem
ID
[16971]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
34.
Power
series
solutions
II:
Generalization
and
theory.
Additional
Exercises.
page
678
Problem
number
:
34.5
(d)
Date
solved
:
Thursday, October 02, 2025 at 01:40:58 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \,{\mathrm e}^{x}&=0 \end{align*}
Using series method with expansion around
\begin{align*} 2 \end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 196
Order:=6;
ode:=sinh(x)*diff(diff(y(x),x),x)+x^2*diff(y(x),x)-exp(x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=2);
\[
y = \left (1+\frac {{\mathrm e}^{2} \operatorname {csch}\left (2\right ) \left (x -2\right )^{2}}{2}+\frac {\left (-8 \,{\mathrm e}^{2}-2\right ) \left (x -2\right )^{3}}{12 \sinh \left (2\right )^{2}}+\frac {\left ({\mathrm e}^{-2}+\frac {33 \,{\mathrm e}^{2}}{2}+\frac {{\mathrm e}^{6}}{2}+12\right ) \left (x -2\right )^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (-392-106 \,{\mathrm e}^{-2}-316 \,{\mathrm e}^{2}-4 \,{\mathrm e}^{-4}-12 \,{\mathrm e}^{4}-10 \,{\mathrm e}^{6}\right ) \left (x -2\right )^{5}}{480 \sinh \left (2\right )^{4}}\right ) y \left (2\right )+\left (x -2-2 \,\operatorname {csch}\left (2\right ) \left (x -2\right )^{2}+\frac {\left ({\mathrm e}^{4}+8 \,{\mathrm e}^{-2}+31\right ) \left (x -2\right )^{3}}{12 \sinh \left (2\right )^{2}}+\frac {\left (-65-7 \cosh \left (4\right )-{\mathrm e}^{2}-47 \,{\mathrm e}^{-2}\right ) \left (x -2\right )^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (410 \,{\mathrm e}^{-4}+{\mathrm e}^{8}+85 \,{\mathrm e}^{2}+68 \,{\mathrm e}^{4}+1537 \,{\mathrm e}^{-2}+11 \,{\mathrm e}^{-6}-{\mathrm e}^{6}+1249\right ) \left (x -2\right )^{5}}{480 \sinh \left (2\right )^{4}}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right )
\]
✓ Mathematica. Time used: 0.024 (sec). Leaf size: 644
ode=Sinh[x]*D[y[x],{x,2}]+x^2*D[y[x],x]-Exp[x]*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
\[
y(x)\to c_1 \left (-\frac {2 e^4 \left (1+4 e^4+e^8\right ) (x-2)^5}{15 \left (e^4-1\right )^4}-\frac {8}{15} e^2 (x-2)^5 \text {csch}^4(2)-\frac {1}{15} e^4 (x-2)^5 \text {csch}^3(2)+\frac {2}{3} e^2 (x-2)^4 \text {csch}^3(2)+\frac {1}{24} e^4 (x-2)^4 \text {csch}^2(2)-\frac {2}{3} e^2 (x-2)^3 \text {csch}^2(2)+\frac {1}{2} e^2 (x-2)^2 \text {csch}(2)+\frac {1}{20} e^2 (x-2)^5 \left (1-4 \coth ^2(2)+4 \coth (2)\right ) \text {csch}^2(2)-\frac {4}{5} e^2 (x-2)^5 (\coth (2)-1) \text {csch}^3(2)-\frac {1}{15} e^2 (x-2)^5 (\coth (2)-1) \coth (2) \text {csch}^2(2)-\frac {1}{10} e^2 (x-2)^5 (\coth (2)-1)^2 \text {csch}^2(2)-\frac {1}{30} e^4 (x-2)^5 (\coth (2)-1) \text {csch}^2(2)+\frac {1}{2} e^2 (x-2)^4 (\coth (2)-1) \text {csch}^2(2)+\frac {1}{12} e^2 (x-2)^4 (\coth (2)-1) \coth (2) \text {csch}(2)-\frac {1}{6} e^2 (x-2)^3 (\coth (2)-1) \text {csch}(2)+1\right )+c_2 \left (x+\frac {32}{15} (x-2)^5 \text {csch}^4(2)+\frac {2}{5} e^2 (x-2)^5 \text {csch}^3(2)-\frac {8}{3} (x-2)^4 \text {csch}^3(2)+\frac {1}{120} e^4 (x-2)^5 \text {csch}^2(2)-\frac {1}{3} e^2 (x-2)^4 \text {csch}^2(2)+\frac {8}{3} (x-2)^3 \text {csch}^2(2)+\frac {1}{6} e^2 (x-2)^3 \text {csch}(2)-2 (x-2)^2 \text {csch}(2)+\frac {1}{15} (x-2)^5 \left (-1+4 \coth ^2(2)-4 \coth (2)\right ) \text {csch}^2(2)-\frac {1}{5} (x-2)^5 \left (1-4 \coth ^2(2)+4 \coth (2)\right ) \text {csch}^2(2)-\frac {1}{12} (x-2)^4 \left (-1+4 \coth ^2(2)-4 \coth (2)\right ) \text {csch}(2)-\frac {1}{60} (x-2)^5 \left (-6-12 \coth ^3(2)+12 \coth ^2(2)+7 \coth (2)\right ) \text {csch}(2)+\frac {16}{5} (x-2)^5 (\coth (2)-1) \text {csch}^3(2)+\frac {2}{5} (x-2)^5 (\coth (2)-1)^2 \text {csch}^2(2)+\frac {3}{10} e^2 (x-2)^5 (\coth (2)-1) \text {csch}^2(2)-2 (x-2)^4 (\coth (2)-1) \text {csch}^2(2)+\frac {1}{20} e^2 (x-2)^5 (\coth (2)-1) \coth (2) \text {csch}(2)-\frac {1}{12} e^2 (x-2)^4 (\coth (2)-1) \text {csch}(2)+\frac {2}{3} (x-2)^3 (\coth (2)-1) \text {csch}(2)-2\right )
\]
✓ Sympy. Time used: 2.534 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), x) - y(x)*exp(x) + sinh(x)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=2,n=6)
\[
y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4}}{12 \sinh {\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4}}{\sinh ^{2}{\left (x + 2 \right )}} - \frac {8 \left (x - 2\right )^{4}}{3 \sinh ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{3} e^{x + 2}}{6 \sinh {\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3}}{3 \sinh {\left (x + 2 \right )}} + \frac {8 \left (x - 2\right )^{3}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{2}}{\sinh {\left (x + 2 \right )}} - 2\right ) + C_{1} \left (- \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{4} e^{2 x + 4}}{24 \sinh ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{2} e^{x + 2}}{2 \sinh {\left (x + 2 \right )}} + 1\right ) + O\left (x^{6}\right )
\]